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Chapter 0: Problem 120

Solve each equation, rounding your answer to four significant digits where necessary. $$ (x-4)^{2 / 3}+1=5 $$

Short Answer

Expert verified
The short answer to the given equation \((x-4)^{2 / 3}+1=5\) is \(x\approx7.635\).

Step by step solution

01

Rewrite the equation

The given equation is: \((x-4)^{2 / 3}+1=5\)
02

Isolate the term containing x

Subtract 1 from both sides of the equation: \((x-4)^{2 / 3}=4\)
03

Remove the fractional exponent

To eliminate the fractional exponent, raise both sides of the equation to the power of the reciprocal of the fraction (in this case, 3/2): \(((x-4)^{2 / 3})^{3/2}=4^{3/2}\) Using the power rule \((a^m)^n = a^{mn}\), we get: \((x-4)^{4/3}=8\)
04

Remove the remaining exponent

Raise both sides of the equation to the power of the reciprocal of the exponent (in this case, 3/4): \(((x-4)^{4/3})^{3/4}=8^{3/4}\) Using the power rule again, we get: \((x-4)=8^{3/4}\)
05

Evaluate the expression and solve for x

Evaluate \(8^{3/4}\) using a calculator to get approximately 3.6349. Now isolate x by adding 4 to both sides of the equation: \(x=3.6349+4\) Finally, add and round the answer to four significant digits if necessary: \(x\approx7.635\) So, the solution is \(x\approx7.635\).

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Most popular questions from this chapter

Calculate each expression in Exencises giving the answer as \(a\) whole mumber or a fraction in lowest terms.\(\frac{12-(1-4)}{2(5-1) \cdot 2-1}\)

Find all possible real solutions of each equation \(x^{3}+4 x^{2}+4 x+3=0\)

By any method, determine all possible real solutions of each equation Check your answers by substitution. \(2 x^{2}-x-3=0\)

Calculate each expression in Exencises giving the answer as \(a\) whole mumber or a fraction in lowest terms.\(3\left(1-\left(-\frac{1}{2}\right)^{2}\right)^{2}+1\)

\((3 x+1)\left(2 x^{2}-x+1\right)\)
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